Last week the Supreme Court took a case that could dramatically scale back or even end affirmative action. In Fisher v. University of Texas, a white woman named Abigail Fisher is suing the University of Texas (UT) for denying her admission using criteria that take race into account. A few years ago, her suit would have been a sure loser — but the court is now less accepting of affirmative action. The question is, How far has it shifted? In 2003, in Grutter v. Bollinger, the court approved a university’s affirmative-action plan. But Sandra Day O’Connor, who provided the critical fifth vote, retired in 2006. Now, all eyes are on Anthony Kennedy, the court’s new swing vote, who could decide affirmative action’s future.

There are several reasons to believe this could be a Big Case. Affirmative-action critics are certainly talking that way. Ward Connerly, president of the American Civil Rights Institute, called the decision to take the case a “potentially historic step.” And it looks like there are at least four Justices ready to take bold action. But there is also reason to believe the court will stop short — weakening affirmative action but not ending it.

The Texas case arises out of UT’s decision to deny Fisher a place in its class of 2008. UT has a two-track admissions system. About 80% of freshmen get in because they graduate in the top 10% of a Texas public high school. The rest are chosen by a formula that takes race into account in a holistic way — not by giving a specific number of points to people of different races.

Fisher is challenging this formula track, arguing that by using race UT deprived her of her 14th Amendment right of equal protection. She lost in the U.S. Court of Appeals for the Fifth Circuit, which makes sense. The Supreme Court has repeatedly said that considering race in a holistic way is O.K. — first in University of California v. Bakke in 1978, and again in Grutter v. Bollinger in 2003. But are the Justices prepared to abandon those precedents and adopt a harder line? Chief Justice John Roberts seems ready to. In a 2007 school integration case, Roberts declared bluntly: “The way to stop discrimination on the basis of race is to stop discriminating on the basis of race.” Justices Antonin Scalia, Clarence Thomas and Samuel Alito are also affirmative-action skeptics. Still, to get a majority, they will need a fifth vote — and that would be Justice Kennedy. In past cases, he has been no great fan of affirmative action, repeatedly voting with the conservative bloc. But he has also written his own opinions, setting out a more nuanced position and has suggested that affirmative action is sometimes warranted, but it must be used very carefully.

So, what is the court likely to do? It could issue a sweeping anti-affirmative-action ruling. More likely, though, Justice Kennedy will put a mild foot on the brakes — saying that most selection systems that consider race (including UT’s) are unconstitutional but that in rare cases taking race into account is acceptable.

On one level, the stakes in Fisher v. Texas are low. Fisher has already attended another college — so the issue for her is not education but money damages. If UT loses, the impact on campus will likely not be huge since the selection method that admits most minority students — the “top 10% track” — will probably not be affected. And since the 14th Amendment applies only to government, the ruling will likely effect only public schools like UT, not private universities.

Despite all this, what the court does could matter a great deal. The nation is deeply divided about affirmative action. Supporters say it has done a great deal to help lift up black and other minority communities. But they say more must be done to ensure that all Americans have a fair chance to succeed. Some affirmative-action critics argue that it has never been fair to penalize young white people like Fisher for problems they had nothing to do with creating. Others argue that there was once a rationale for affirmative action, but it is now time to put it aside and look beyond race.

If the Supreme Court strikes down UT’s admissions system, other public schools will have to re-evaluate their admissions policies — and other institutions, from private schools to employers, will likely rethink how they consider race. Affirmative action will not disappear overnight, but the Supreme Court’s conservative bloc appears to be intent on reducing the role it plays in university admissions — and other parts of society.